Imaging quality improvement for wavefront colding system with radially symmetrical phase mask

In this paper, we introduce the novel method to improve the image quality for the optical imaging system with the radially symmetrical phase masks. By introducing 0-π phase mask into the radially symmetrical one, the new phase mask is generated, which produces the dark-shape point spread function. By rotating this phase mask with the amplitude mask for different angles, the point spread function is still rotated by the same angle. Moreover, the digital processing is used to restore the high-quality image. The simulation results demonstrated the effectiveness of the proposed method in comparison to the previously radially symmetrical phase mask.


Introduction
Due to limit of the light diffraction, the optical imaging system has the spatial resolution and the depth of filed for the given number aperture (NA) and wavelength (Saavedra et al. 2009;Le 2023). The depth of filed shows the imaging ability of optical system along optical axis. When the depth of field is big, much information of three-dimensional objects will be acquired. Recently, wavefront coding technique has been a powerful tool which is used to extend the depth of field (Muyo et al. 2009). In wavefront coding technique, in order to control the invariant of the point spread function to defocus, a phase mask will be placed in the pupil plane. There are two types of phase mask developed to control the point spread function (PSF), which are the radially symmetrical phase mask and asymmetrical phase mask. There are some asymmetrical phase masks introduced to control the point spread function, such as the cubic phase mask (Dowski and Cathey 1995), the tangent phase mask (Le et al. 2014), the logarithm phase mask (Zhao and Li 2010a, b), the square roof phase mask , the sinusoidal phase mask (Zhao and Li 2010a, b), metasurfaces (Zhang et al. 2021), freeform (Moein and Suleski 2021). There are some radially symmetrical phase masks suggested to extend the depth of field, such as quartic phase mask (QPM) (Nhu et al. 2016), logarithmic axicon (Sochacki et al. 1992), diffraction hybrid lens (Zalvidea and Sicre 1998), and logarithmic asphere (Chi and George 2001). The radially symmetrical phase mask generates the radially symmetrical point-spread-function. The PSF is the sharper and small, so the images captured camera have the accepted quality. Meanwhile, the asymmetrical phase masks reach the large size of the point spread function so the images captured camera are usually low-quality. With a view to obtaining sharper images, the digital processing is carried out to restore images. However, there raises a problem that the image artifacts often appear on the restored image.
The cutoff frequency of the asymmetrical phase masks is near set to that of the traditional optical system. In the meantime, the cutoff frequency of the radially symmetrical phase mask is smaller than that of the traditional optical system. In addition, the MTF value of the radially symmetrical phase mask is relatively low in comparison to the diffraction-limit MTF value of the traditional optical system. Therefore, the imaging quality of the radially symmetrical phase mask is relatively lower than that of the optical system at in-focus plane. In order to enhance the imaging quality of the radially symmetrical phase masks, in this article, we introduce the novel way to improve them so that generates the new point spread function. By introducing the 0-π phase mask into the radially symmetrical phase mask, the dark-shape point spread function will be obtained. The dark-shape point spread function will elevate the MTF value in the big frequency region which means that the imaging quality will be advanced. By rotating this phase mask in different angle, the MTF value in the big frequency region will be enhanced in the responding direction. When using RL algorithm to restore the image, the higher image quality will be acquired.
In this paper, we present the novel method based on the usage of rotating the improved mask with the four directions to obtain the invariant and high-quality images over a wide range of defocus. Section 2 shows the imaging theory of the proposed method. Section 3 illustrates the simulation results and analysis. Finally, the conclusions are presented in Sect. 4.

Imaging theory
The recorded image can be presented by: where o is the object; h is the point spread function; ⊗ is the convolution operator. The point spread function, h, can be presented by: where p is the pupil function; x and y are the normal coordinates in the pupil plane.
The pupil function of the imaging system can be represented by: where A is the amplitude mask, f is the phase mask, ψ is the defocus. The development of the radially symmetrical phase masks to extend the depth of field produces somes phase masks such as:. In this paper, we use the tangent phase mask and the quartic phase mask to perharpe the ability of the improvement of the proposed method. The quartic phase mask is the most commotical phase mask used researched for the radially symmetrical phase mask. Meanwhile, the tagent phase mask is the one to the near best invariant of the modulation transfer function (MTF) to defocus.
The quartic phase mask can be presented by : To obtain the optimal imaging quality for optical system with phase masks, the phase mask parameters should be optimized. In this paper, we used the optimal phase mask parameters which are shown in the previous papers . They are equal to a = 17.65, b = 14.6. The profile of the quartic phase mask is shown in Fig. 1(a). The point spread function of the quartic phase mask is shown in Fig. 1(b). Figure 2 is the spokes sample which is used to the imaging simulation processing. Figure 3 shows the images of the conventional imaging system and the quartic phase mask for different values of defocus. The images of the conventional imaging system are shown the top of Fig. 3. The images of the quartic phase mask are shown the bottom of Fig. 3. For the defocus value of ψ = 0, the higher image of the conventional imaging system is achieved. Nevertheless, the defocus value is increasing, the imaging quality of the conventional imaging system is fast reduced. As can be seen in Fig. 3(c), the edge of the spokes is blurred and the phase shift appears, creating a low-quality image. The images of the quartic phase mask is lower than the image of the conventional imaging system at defocus value of ψ = 0. However, the image quality of the quartic phase mask is near invariant over big range of defocus from ψ = 0 to ψ = 10. The edge of the spokes for the images of the quartic is till sharp.
The improved pupil can be writted by:    and AM improved (x, y) = x 2 ifx 2 ≤ 1 0 other (5) The o-π phase mask as shown in Fig, 4(c) is added into the quartic phase mask, which produces the improved phase mask. Moreover, the amplitude mask will be added. The amplitude mask is shown in Fig. 4(a) and the improved phase mask is shown in Fig. 4(b).
The MTFs of the quartic phase mask and the improved one are shown in Fig. 5(a) and Fig. 5(b), respectively. From Fig. 5 it is clear that the MTF value of big frequency region of the improved mask is higher than that of the quartic phase mask.

Simulation result
The setup of the optical imaging system is shown in Fig. 6. The amplitude and phase masks mounted on the mechanical assembly are illustrated in Fig. 6. By rotating this mechanical body, the amplitude and phase mask will be rotated by the same angle. Thus, the point spread function will rotate by the same angle while Fig. 7 shows it with the responding angles. As illustrated in Fig. 7, when the mechanical body rotates, the point spread function will rotate the same angle at ψ = 0.
In this article, we use the blind post processing to achieve the high-resolution image. As can be seen in Ref. (Richardson 1972), the blind post processing can correct the effect of the optical aberrations. After that, we chose the Richardson-Lucy deconvolution to achieve the blind post processing, and the iterative formula for an image can be presented by.
where o is the object; g is the input image; h is the system point spread function; h r is the flipped point spread function; t is the number of iterations; ⊗ is the convolution operator.
Since there are four raw images, the iterative process can be presented by, With a view to showing the imaging effectiveness, we perform the comparison of the proposed method with the imaging system with the quartic phase mask. For the digital processing, the starting parameters are: o 1 = norm(g 1 + g 2 + g 3 +g 4 ); h 1 , h 2 , h 3 and h 4 are the point spread functions at ψ = 0 for four different angels. Based on the starting conditions, the simulation results are shown in Fig. 8 where the images of the quartic phase mask and the proposed method are illustrated. The proposed method is shown on top of Fig. 8. The image of the quartic phase mask is shown in bottom of Fig. 8. From Fig. 8, the edge of the spokes images of the proposed method are sharper than that of the quartic phase mask at all defocus values.
For the purpose of measuring the imaging quality of the images in Fig. 8, we use the structural similarity index measure function (SSIM). SSIM is used for measuring the similarity between two images. The SSIM value is from 0 to 1. The bigger SSIM value, the higher image quality. The SSIM values for the images of the quartic phase mask and the proposed method on Fig. 8 are shown on Table 1. The SSIM values for the images of the quartic phase mask at ψ = 0, 5 and 10 are set to SSIM = 0.6726, 0.6351 and 0.5670. It can be seen that the bigger defocus value, the smaller SSIM value. The SSIM values for the images of the proposed method at ψ = 0, 5 and 10 are set to SSIM = 0.8888, 0.8290 and 0.7043. It is clear that the SSIM value for the images of the proposed method is bigger than that of the quartic phase mask at each defocus value. This means that the images of the proposed method have higher quality.
Afterwards, we consider the noise effectiveness to the imaging quality for the traditional system and proposed method. We add the white Gaussian noise on the image simulations at  method are equal to 0.5473 and 0.7202, respectively. For SNR = 20dB, the SSIM values of the quartic phase mask and the proposed method are equal to 0.3018 and 0.5646, accordingly. That means the proposed method has better imaging quality.
Finally, we test with different original images at the defocus value of ψ = 0. The original images are shown on the left in Fig. 10. The image simulation for the quartic phase mask and the proposed method are shown in Fig. 10. The pictures in the middle column in Fig. 10 are illustrated as the images of the quartic phase mask. The pictures in the right column in Fig. 10 are indicated the images of the proposed method. From Fig. 10, it is not difficult to see that the images of the proposed method are sharper than that of the quartic phase mask. In order to show clearly the image quality with different ways, we use SSIM function. The SSIM values of the images of the quartic phase mask and the proposed method are shown in Fig. 10. The SSIM values of the images of the proposed method are bigger than that of the quartic phase mask for each original image. Hence, it can be inferred that the images using proposed method have higher quality than those using quartic phase mask.

Conclusion
We have been successfully suggested a novel method to obtain higher quality image for optical imaging sytem with radially phase mask. A quartic phase mask is used to demonstrate the effectiveness of the proposed method. The imaging results of the quartic phase mask and the proposed method are shown. The simulation results demonstrated that the proposed method can be employed to improve the imaging quality for the radially symmetrical phase masks.
Authors' contributions Vannhu Le mainly writes the manuscript. Minhthai Le corrects the manuscript. All authors review the manuscript.
Funding not applicable.

Declarations
Ethical approval Not applicable.